AISB'01 Invited Talk - Christoph Benzmüller (23rd March)

An Agent Based Approach to Reasoning

In this talk I shall present an agent based approach to automated and
interactive reasoning. The approach is motivated by the discrepancy of
the flexible problem solving behavior of human mathematicians on the
one hand and the control determination of automated reasoning systems
on the other hand. Depending on the specific nature of a challenging
problem in real mathematics or engineering different specialists may
have to cooperate and bring in their expertise to effectively tackle a
problem. Even a single mathematician possesses a large repertoire of
often very specialized reasoning and problem solving techniques. But
instead of applying them in a fixed structure, a mathematician uses
own experience and intuition to flexibly combine them in an
appropriate way.

Although classical automated theorem provers and proof planners have
reached an impressive power and do already outperform humans in
particular problem domains, these systems often perform poorly when
applied to new domains. Especially proof planning has a problem if
accurate method and/or control knowledge for the new domain is
lacking. Cross domain problems also provide a good example as they
typically require the combination of quite heterogeneous reasoning
techniques. In extreme cases the required spectrum reaches from
different theorem proving strategies to symbolic computation or model
generation.

A recent trend in the automated reasoning community therefore is to
integrate specialized external reasoning systems in a central theorem
proving environment. However, in traditional theorem proving as well
as proof planning external reasoning systems are integrated in a quite
hard-wired fashion making a flexible, reactive interplay impossible.
Instead the control mechanismen of the central prover typically
precisely determines when external systems are applied and how they
cooperate. Furthermore, the adaptation of these static control
mechanisms at run-time is rarely possible. As a consequence, for
instance, proof subgoals that could be automatically solved by an
integrated external reasoner within seconds may remain undetected
simply because appropriate control knowledge is not available in the
core system. This clearly contradicts with the intuitive or
unconscious handling of different proof strategies of human
mathematicians.

In contrast to hard-wired integrations our approach provides a
flexible integration framework for machine-oriented theorem proving,
tactical theorem proving, proof planning, and computation. It
encapsulates proof rules, tactics, methods, and external systems in
single reasoning agents and uses state of the art distribution
techniques to decentralize and spread them over the internet. The
approach particularly supports cooperative proofs between reasoning
agents which are strong in different applications areas, e.g.,
higher-order and first-order theorem provers, and computer algebra
systems. Nevertheless, the integration approach is a skeptical one
and assumes that the contributions of external systems can be
translated in a central proof object (based on a higher-order variant
of Gentzen's natural deduction calculus), where they can be verified
at fully expanded calculus level or investigated by the user at a more
user friendly abstract level.

Our approach particularly allows a number of heterogeneous proof
search attempts to be executed in parallel. This includes tackling a
goal with different integrated theorem provers as well as checking for
counterexamples with an integrated model generator. If these reactive
proof attempts fail, the integrated reasoners may cooperate by
exchanging interesting partial results via the central proof object.
Resource management is employed to distribute available resources
amongst the available reasoning agents in order to determine their
maximum reasoning time and to prevent them from a quick consumption of
all available resources. Generally our approach supports
parallelization of reasoning tasks on term level, inference level, and
proof search level and external systems can not only be employed to
support object level proof search but also to support meta-level
reasoning, for instance, by checking complicate application conditions
of inference rules.

Additional advantages of our approach consist in the close integration
of automation and interaction and the potentiality to add, delete, or
modify reasoning agents and control information at run-time. These
features are based on the special characteristics of the Omega-ants
blackboard mechanism forming the core of our system. The task of the
Omega-ants agents (the knowledge sources of the blackboards)
is to test the applicability of rules, tactics, methods, and external
reasoners in a given proof state and to suggest appropriate parameter
instantiations for them. The resulting suggestions are offered for
execution to the interactive user and the automated prover
simultaneously.

The system has been successfully applied to different problems about
sets and currently also to prove the equivalence of different group
definitions. The set examples are successfully classified in valid and
invalid statements in a reactive interplay of natural deduction proof
search, higher-order and first-order resolution style theorem proving,
computation, and refutation with a model generator. Thereby the
system demonstrated a resource adapted proof search behavior.
However, the main bottleneck for obtaining large proofs is the
translation between the different systems involved, in particular, of
large clause sets.

The system is realized in the Saarbrücken Omega proof
environment, and implemented in concurrent, object-oriented
Lisp. However, conceptually it does not depend on this framework. We
belief that it is even independent of the theorem proving context and
may be adapted to other rule based reasoning systems.

The agent-based reasoning framework is joint work with my colleagues
Volker Sorge (Saarland University), Manfred Kerber, and Mateja Jamnik
(The University of Birmingham).

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